Why doesn’t a/b=(a+x)/(b+x)?
Conceptually if you are increasing both numbers a and b by an equal amount I don’t understand why the fraction would not also keep the same value. I would appreciate the clarification.
- Bulldog reduxLv 74 weeks ago
Here's a refutation:
2/4 = .5
(2+3)/4+3) = 5/7 = .714
- charlatanLv 74 weeks ago
a/b = m/1
(a/b)+p = m+p
(a+p)/(b+p) = (m+p)/(1+p)
any action or operation like +,-,*,/ on one side of (=) sign,
should also be the other side,then only resulting
ratio will remain same
if you divide one side values, the values of the other side
quotient will be always = 1
- Jeff AaronLv 74 weeks ago
ab = (a + x) / (b + x)
a(b + x) = b(a + x)
ab + ax = ab + bx
ax = bx
True only if a = b or x = 0
So ab = (a + x) / (b + x) is sometimes true, but not always. For example:
7/7 = (7 + 5) / (7 + 5): TRUE because a = b
3/4 = (3 + 0) / (4 + 0): TRUE because x = 0
5/7 = (5 + 2) / (7 + 2), i.e. 5/7 = 7/9: FALSE
Note that b can't be zero and b + x can't be zero.
- davidLv 74 weeks ago
increasing by addition is NOT the same as increasing by multiplication
2 + 3 = 5 while 2 X 3 = 6 . . why .. because multiplication is repeated addition. 2X3 means 2 + 2 + 2 = 6
similarly division decreases something but not in the same way subtraction does . . .
[a + (1/2)a] / [b + (1/2)b] does equal a/b
similarly [ (a + xa)/(b + xb)] = a/b
... these are examples of proportionally increasing both the numerator and the denominator,,,
I suppose that the answer to your question comes from ancient math. 5000 years ago they had no rules, Repeated observations by many people over time led to the accepted rules of algebra.
let's try your proposition with real numbers
1/2 = (1 + 3)/(2+3) ???
. . . .right side becomes 4/5
so does 1/2 = 4/5? NO so this cannot be a rule of math,
. . . try it with any numbers that you want (except adding 0) and it does not form an equality. That is why this is not allowed in math.
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- PuzzlingLv 74 weeks ago
In order to keep the ratio the same, you must *multiply* (or divide) both numbers by the same value.
In other words:
a/b = ak/(bk), where k is some non-zero constant.
In fact, when you are reducing fractions, you are looking for that common factor between the numerator and denominator. You then divide each by that number and you get an equivalent reduced fraction.
8/12 = 2/3
Here you divide top and bottom by 4. Or stated in reverse, if you had the fraction 2/3, you could get an equivalent fraction of 8/12 by multiplying top and bottom by 4.
Here's one more way to think about it.
You have a fraction:
The only thing you can multiply that by and have it still equal the same value is 1.
a/b * 1 = a/b
So let's convert 1 into an equivalent fraction of k/k (assuming k isn't 0):
a/b * k/k = a/b
(ak)/(bk) = a/b
See, you have to *multiply* top and bottom by the same value to keep the fraction the same. You can't just *add* the same amount.
- 4 weeks ago
nononoo. a/b is a proportion. Therefore anything that you add on the top and bottom has to be scaled to that proportion for it to be equal. The only way this could be true is if a = b.
- roderick_youngLv 74 weeks ago
When doing sums or differences, this works.
When doing products or quotients, you need to increase by the same percentage to be considered "the same amount." For example, with a/b , if you increase both a and b by 27%, the fraction will still have the same value.
- lenpol7Lv 74 weeks ago
Try a couple of numbers:-
a/b = > 5/10 = 1/2 ( by cancelling down ).
(5 + 2) / (10 + 2) = 7/12 ( This is neither half nor cancels down).
NB My answer is NOT a proof, but just a numerical verification.
hope that helps!!!!
- Keith ALv 64 weeks ago
Take a simple case:
a = 0 , b = 1 ; and x any integer.
E.g., (0 + 2) / (1 + 2) = 2/3 , which is not = 0 .
(Again, by example:)
If you divide a pizza into three parts and take one, that is clearly not the same as dividing it into five parts and taking three.
- Anonymous4 weeks ago
I decided to visualize this geometrically with an example (see photo) and it really clicked! Thank you for your quick responses. The reason I had this question is bc a ratio of T/T0 only works when the units are in kelvin (unit: 273+C) and not in Celsius (C)
which perplexed me! Turns out this is the reason why